Commensurated subgroups, semistability and simple connectivity at infinity

TL;DR

This paper explores the properties of finitely generated groups with commensurated subgroups, establishing conditions under which such groups are semistable or simply connected at infinity, thus extending classic geometric group theory results.

Contribution

It generalizes key results on semistability and simple connectivity at infinity for finitely presented groups involving commensurated subgroups.

Findings

Groups with infinite finitely generated commensurated subgroups are semistable at infinity.

Under certain conditions, such groups are also simply connected at infinity.

Provides a short proof of Lew's theorem on subnormal subgroups and semistability.

Abstract

A subgroup Q is commensurated in a group G if each G conjugate of Q intersects Q in a group that has finite index in both Q and the conjugate. So commensurated subgroups are similar to normal subgroups. Semistability and simple connectivity at infinity are geometric asymptotic properties of finitely presented groups. In this paper we generalize several of the classic semistability and simple connectivity at infinity results for finitely presented groups. In particular, we show that if a finitely generated group G contains an infinite finitely generated commensurated subgroup Q of infinite index in G, then G is semistable at infinity. If additionally G and Q are finitely presented and either Q is 1-ended or the pair (G,Q) has one filtered end, then G is simply connected at infinity. This result leads to a relatively short proof of V. M. Lew's theorem that finitely presented groups with infinite finitely generated subnormal subgroups of infinite index are semistable at infinity.